Short-Term Wind Power Prediction Based on a Hybrid Markov-Based PSO-BP Neural Network

Abstract

Wind power prediction is an important research topic in the wind power industry and many prediction algorithms have recently been studied for the sake of achieving the goal of improving the accuracy of short-term forecasting in an effective way. To tackle the issue of generating a huge transition matrix in the traditional Markov model, this paper introduces a real-time forecasting method that reduces the required calculation time and memory space without compromising the prediction accuracy of the original model. This method is capable of obtaining the state probability interval distribution for the next moment through real-time calculation while preserving the accuracy of the original model. Furthermore, the proposed Markov-based Back Propagation (BP) neural network was optimized using the Particle Swarm Optimization (PSO) algorithm in order to effectively improve the prediction approach with an improved PSO-BP neural network. Compared with traditional methods, the computing time of our improved algorithm increases linearly, instead of growing exponentially. Additionally, the optimized Markov-based PSO-BP neural network produced a better predictive effect. We observed that the Mean Absolute Percentage Error (MAPE) and Mean Absolute Error (MAE) of the prediction model were 12.7% and 179.26, respectively; compared with the existing methods, this model generates more accurate prediction results.

1. Introduction

With increasing fossil fuel depletion and the carbon neutrality target in China, clean energy has a great potential for fundamentally solving the resource constraints and environmental pollution problems facing the human energy supply and achieving sustainable energy development [1]. Renewable energy is an effective way to bring about social transformation involving the green economy [2,3]. The Global Wind Energy Report 2022, published by the Global Wind Energy Council (GWEC) [4], shows that installed wind power capacity has been increasing since 2010, as shown in Figure 1. Wind power has grown rapidly over the past few decades, with installed wind power capacity increasing from 39 GW a year in 2010 to 93.6 GW a year in 2021, reaching a total capacity of 837 GW. An estimated 55.9 GW of new Chinese wind capacity (41.4 GW onshore and 14.5 GW offshore) will be added to the country’s total installed capacity of 346.7 GW, of which about 47.6 GW will be added to the National Grid, of which the majority (61%) are in the more populous central, eastern, and southern regions.

However, the grid-connected output power in wind farms is affected by wind changes and its behavior is extremely unstable. Ensuring the internal balance and smooth operation of the wind system is an important factor for increasing the power of wind turbines. The variation of wind power has a binding effect on the degree of the power utilization of generators, so the accurate prediction of wind farm power becomes an important task. At the same time, as an effective measure for realizing optimal control of wind power, wind power prediction is very important for guiding system scheduling operations and wind farm production arrangements.

Among the traditional statistical forecasting models for wind power, Markov-based prediction models are prominent in terms of their performance. This prediction model is simple and practical and can simulate wind time-series correlation properties, resulting in a considerable amount of research in this area [5,6,7,8,9,10,11,12].

In terms of simulating time series, in [5,6] they studied variation patterns in wind speed using first-order and second-order Markov models and demonstrated that the model could truly reflect the statistical properties of the original data. Papaefthymiou and Klöckl [7], on the other hand, illustrated that the direct generation of wind power time series for the development of synthetic models can reduce model complexity while generating good simulation results by experimenting with first-order, second-order, and third-order Markov models. Xie et al. [8] proposed a Non-Homogeneous Markov Chain (NHMC) wind speed model that takes into account the daily and seasonal characteristics of wind speed variations.

D’Amico et al. proposed three semi-Markov models using time-lag autocorrelation to compare the statistical properties of the proposed model with those of real data and their model effects outperformed the simple Markov model in terms of the statistical properties of wind speed data [9]. Yousuf et al. proposed a prediction model that combines MCs with tuned dynamic moving windows, though its results are similar to other neural network-like models [10]. Zhao et al. proposed an ultra-short-term WPF spatiotemporal Markov chain model by extending the traditional discrete-time Markov chain with off-site reference information to improve the prediction accuracy of regional wind farms [11]. Yang et al. considered that the selection of weights in traditional weighted Markov models is more dependent on the initial conditions, and the application of fuzzy data sets to determine the weights could improve the accuracy of power predictions [13].

Markov models have been applied in other fields as well [14,15,16,17,18,19,20,21,22,23,24,25]. For example, Munkhammar et al. used the Markov Chain mixed distribution Model (MCM) for very short-term load forecasting of residential electricity consumption [14]. Munkhammar et al. made Markov chain predictions about solar irradiance [15,16]. Guan H. et al. proposed a forecasting model combining Markov chain theory and fuzzy set theory to reveal the inherent volatility patterns hidden in stock market time-series data [17]. Li W. et al. used information from wind acceleration to improve wind speed prediction accuracy based on a Markov model [18].

Although the traditional MC model shows better performance than other benchmark models [26], the method is computationally intensive and is limited in its performance over a large range of datasets. A major reason is its ineffective state classification method. The conventional state classification implemented by Shamshad et al. generated huge Transfer Probability Matrices (TPM), which increased the computational complexity [5,6,7,26]. In higher-order classification states, most of the elements of the matrix are zero, and the valid values are mainly concentrated on the main diagonal. H. Yang et al. used a method which reduced the number of states to overcome the problem of oversized TPM, but prediction accuracy was affected [27].

Based on the above discussion, we find the following two issues within the traditional Markov-based prediction approaches model:

• State classification of higher-order Markov models: If the interval of the model is divided into too many parts, a huge TPM is generated, which leads to an increase in computational complexity. If the division is too small, prediction accuracy will be affected. In addition, higher-order models that produce TPMs may be difficult to even store.
• The point forecast model needs further improvement: In the literature authored by Carpinone et al., some Markov models use probability intervals for prediction [28], some base the final prediction on weighting [11,13], and some use the mean of the maximum probability interval for prediction [29]. How to determine the state interval at the next moment and estimate its final point prediction value still need further research.

Therefore, we consider how to design a real-time method to calculate the next-time state probability interval of the Markov model so as to optimize computational complexity while ensuring the accuracy of the prediction is consistent with the original model. Through the exploration of different point forecasting methods, we seek to improve the accuracy of wind power forecasting.

The main contributions of this paper are summarized as follows:

• For the first problem, we study the idea of building a Markov model and using a real-time calculation to obtain the state probability interval of the next moment of the model, which can be applied to both wind power prediction and generate wind speed data.
• For the second problem, we propose a combined model that combines the Markov model with a Particle Swarm Algorithm (PSO) and Back Propagation (BP) neural network for short-term wind power prediction. In addition, we compared the first-order Markov model, the second-order Markov model, the third-order Markov model, and the weighted Markov model for point prediction.

This paper is structured as follows. In Section 2, we introduce the Markov-based prediction models and improved algorithms for higher-order Markov processes. In Section 3, we present a Markov-based prediction approach combined with BP neural networks and a PSO algorithm, and five performance indicators are defined for evaluating the prediction algorithms. Next, in Section 4, we illustrate the experimental results based on two data sets. Finally, our conclusions are summarized in Section 5.

2. Markov-Based Prediction Approach

In this study, a combined Markov-PSO-BP neural network model was used to predict wind power. The research focuses on model improvement, comparative performance studies, and reduction of CPU Time operations for higher-order Markov models. Comparing the first-order Markov model, the second-order Markov model, the third-order Markov model, and the weighted Markov model under different segmentation intervals, we selected a combination of first-order Markov models with BP neural networks and PSO which uses the Mean Absolute Error (MAE), Root Mean Square Error (RMSE), Mean Square Error (MSE), Mean Absolute Percentage Error (MAPE), and the Coefficient of Determination (R-squared) as evaluation metrics for forecasting. We also investigated the problem of excessive Transfer Probability Matrices for higher-order Markov models and proposed a real-time computation method that reduces the computation time for high-dimensional matrices in higher-order Markov models.

2.1. Single-Order and Multi-Order Markov Models

The Markov model used in this study, concerning [30], reconciles the model based on a single observed data point. In this study, the data points are generated at the one-time step and the model makes a point forecast of the generation at the next time step.

A Markov model is a stochastic process with a discrete time and state, defined as follows: If there exists such a stochastic process, $\{X_t,t\in T\}$, satisfying a discrete time series $T=\{t=1,2,\dots ,n,\dots \}$, the system state space is $S=\{s_1,s_2,\dots ,s_m,\dots \}$ with the following relationship:

$$P\{X_n=s_n|X_{n-1}=s_{n-1},\dots ,X_2=s_2,X_1=s_1\}=P\{X_n=s_n|X_{n-1}=s_{n-1}\}$$

(1)

then it is said to be a Markov model. Equation (1) indicates that the state $X_n=s_n$ at $t=n$ has nothing to do with the state before the moment $t=n-1$, but only with regards to the state $X_{n-1}=s_{n-1}$ at the moment $t=n-1$; that is, to meet the “no posteriority” requirement.

When the initial moment $t=n$ is in the state of $X_n=s_n$ according to Markov’s “no posteriority” principle, the probability of transferring the system from the state $X_n=s_n$ to $X_{n+p}=s_{n+p}$ at the moment of $n+p$ can be calculated via the following equation:

$$P\{X_{n+p}=s_{n+p}|X_n=s_n\}=P\{X_{n+p}=j|X_n=i\}=p_{ij}(n,n+p),(i,j\in S)$$

(2)

If the transfer probability $p_{ij}(n,n+p)$ satisfies the following conditions, meaning when it is only related to its own position in the matrix and the time duration of the difference between the two moments, we write it as $p_{ij}(p)$. When $p_{ij}(n,n+p)=p_{ij}(p)$, we consider this Markov to be homogeneous or time-homogeneous and consider the transfer probability to have smoothness. If the state space $S=\{s_1,s_2,\dots ,s_n,\dots \}$ is not an infinite set, but is finite in the range of real numbers, then $\left\{X_t,t\in T\right\}$ is said to be a finite-state Markov. In the homogenous Markov state, the state transfer probability can be defined as:

$$p_{ij}(p)=P\{X_{n+p}=j|X_n=i\}$$

(3)

The transfer probability matrix at this point is called the p-step transfer probability.

If the number of system states is $m$, i.e., the system state $S$ can be described as $S=\{s_1,s_2,\dots ,s_m\}$, Markov’s one-step state transfer probability matrix can be represented with the following structure:

$$P=\left[\begin{array}{cccc}{p}_{11}& p_{12}& \dots & p_{1m}\\ p_{21}& p_{22}& \dots & p_{2m}\\ ⋮& ⋮& \ddots & ⋮\\ p_{m1}& p_{m2}& \dots & p_{mm}\end{array}\right]$$

(4)

Each row in the matrix of Equation (4) represents the set of probabilities that a state will be transferred to the next state after one step at a given moment. For example, the probability of state $X_t=s_i$ transferring to state $X_{t+1}=s_j$ at moment $t$ is $p_{ij}$. Assuming that the distribution vector in the initial state is $P_0$, then the probability of the state after the p step can be calculated via the following equation:

$$P(p)=P_0{P}^p$$

(5)

The initial probability vector is calculated from the initial data and combined with the calculated one-step transfer probability matrix; thus, the probability distribution for the moment that needs to be predicted can be derived.

The resulting Markov model prediction steps are described as follows:

• Acquisition of operational data from actual wind farms and pre-processing of the selected data;
• Slice the preprocessed data into m intervals;
• Statistical calculation of the state sequence to obtain the transfer frequency matrix and the one-step transfer probability matrix;
• The initial probability vector is calculated from the initial data and combined with the calculated one-step transfer probability matrix, and thus the probability distribution for the moment to be predicted can be derived;
• The average of the probability interval with the highest probability in the probability distribution is taken as the state of the next moment.

It is important to emphasize that if the number of partition intervals is too large, there may be states with zero transition probability to the next state, resulting in the model’s inability to estimate the probability distribution of the next time step at these states. Although this issue did not occur in this study, it is recommended that one use the true value of the previous step in such cases, since changes in wind power are generally continuous and do not vary much over short periods.

Furthermore, there may be multiple maximum probabilities for the next time step, and in such cases, we compare against the state at the previous moment and select the closest of them. This approach is justified because wind power does not change much over short periods.

2.2. Improved Multi-Order Markov-Based Real-Time Algorithm

In previous studies, an inherent drawback of higher-order Markov models has been the existence of a large Transfer Probability Matrix, which greatly increases computational complexity [1,2,3,16]. In response, we propose a real-time computational approach for avoiding the generation of huge transfer probability matrices while ensuring that their predictions are consistent with those of traditional models.

The detailed steps for improving the k-order Markov algorithm are as follows:

• Enter the pre-processed wind power time series $\{P_1,P_2,\dots ,P_n\}$, number of segments $m$, training points $trian\_num$, prediction points $pre\_num$, and Markov model order $k$ ($k\ge 2$);
• The pre-processed data is sliced into m intervals with interval sizes of $markov\_gap=(P_{\max }-P_{\min })/m$;
• Assign $\{P_1,P_2,\dots ,P_{train\_num}\}$ to the corresponding interval and obtain the sequence of the wind power state interval $P^g=\{P_1^g,P_2^g,\dots ,P_{train\_num}^g\}$;
• The sequences of Markov state transfers satisfying order k in the sequence of wind power intervals are found via retrieval and the values of their next positions are counted;
• The value with the highest frequency in the statistics is used as the predicted value. Let $t=t+1$ be the initial state at the next moment and repeat step 4 times;
• Satisfy $t>pre\_num$ to end the loop and return the final prediction sequence.

The pseudo-code of our improved k-order Markov-based real-time algorithm is given as Algorithm 1.

Algorithm 1 Improved k-order Markov-based real-time algorithm.

Input: After pre-processing, the wind power time series $\{P_1,P_2,\dots ,P_n\}$, number of segments $m$, training points $trian\_num$, prediction Points $pre\_num$, and Markov model order $k$$(k\ge 2$). Output: Markov Model Predicted Power $P_t^{out}$. 1: $markov\_gap=(P_{\max }-P_{\min })/m$ 2: for $i=1,\dots ,train\_num$ do 3:       $P_i^g=[P_i/markov\_gap]$ where [$x$] is the rounding function 4: end for 5: for $t=1,\dots ,pre\_num$ do 6:         $\{P_1^{one},P_2^{one},\dots ,P_j^{one}\}=Find(\{P_1^g,P_2^g,\dots ,P_{train\_num}^g\}=[P_t^{pre}/markov\_gap])$ 7:        for $i=1,\dots ,k$ do 8:               $\{P_1^i,P_2^i,\dots ,P_s^i\}=Find(P_{P^{trains}+1}^g=[P_{t+i}^{pre}/markov\_gap])$ 9:               $P^{trans}=\{P_1^k,P_2^k,\dots ,P_s^k\}$ 10:        end for 11:       $P_t^{out}=P_b^k\times markov\_gap+0.5\times markov\_gap$ 12: end for

A flow chart of the improved k-order Markov-based real-time algorithm is shown in Figure 2.

In the improved model, instead of generating both the transfer frequency matrix and the transfer probability matrix, we retrieve the training data set to determine the interval with the highest probability for the next time step. This approach reduces the computational complexity of the model and avoids the potential problem of having insufficient memory for computation which may arise in higher dimensions while also facilitating the computation of k-order Markov models. However, this algorithm is limited by the fact that an increase in the number of prediction points may lead to longer CPU Times due to the real-time computation, which we verified in our subsequent experiments.

2.3. Weighted Markov-Based Prediction Model

The weighted Markov chain is based on the Markov chain, and according to the different magnitude of the influence of the state transfer probability matrix of each step on the predicted value, the weights are obtained based on the autocorrelation coefficients of each step, and then the final predicted value is obtained by weighting the predicted results of each step [31].

The steps of the weighted Markov chain are as follows:

• Set the criteria for grading;
• Distinguish the state corresponding to each data point;
• Calculate the autocorrelation coefficients of different orders according to the following Equation:

$$r_p=\frac{{\displaystyle \sum _{i=1}^{train\_num-p}(x_i-\overline{x})(x_{i+p}-\overline{x})}}{{\displaystyle \sum _{i=1}^{train\_num}{(x_i-\overline{x})}^2}}$$

(6)

4.

Normalize the autocorrelation coefficients of the different orders according to the following Equation:

$${\omega }_p=\frac{\left|r_p\right|}{{\displaystyle \sum \left|r_p\right|}}$$

(7)

5.

The results of the previous step are counted to obtain the transfer probability matrix at different step sizes;

6.

Assuming that the previous data are the initial states, the state probabilities can be found using the corresponding transfer probability matrix;

7.

Let the weighted sum of the probabilities of each state in the same state be the predicted probability; repeat steps 1 through 6 to perform the next round of prediction.

3. Markov-Based Prediction Approach Combined with the PSO-BP Neural Network

3.1. Hybrid Model

3.1.1. Markov and BP Neural Networks

BP neural networks are one of the most common artificial neural networks, developed by Rumelhart and McClelland in 1986 [32]. Their core mathematical tool is the chain derivative rule of calculus. Using the idea of gradient descent and the gradient search technique, the value of the loss function minimizes the actual output value of the network and the desired output value where the loss function is the function used to measure the difference between the predicted and true values during the supervised learning process. In short-term wind power prediction modeling, three-layer BP neural networks are generally used for reasons such as training speed and data adaptability.

The BP neural network model consists of an input layer, a hidden layer, and an output layer; each layer consists of some neuron nodes. The upper layer and lower layer nodes are connected by the connection weights. When a pair of inputs is given to the input neurons, the neuron activation values are passed from the input layer through the hidden layer to the output layer, and then the error between the network output and the actual output samples is reduced according to the direction from the output layer, the reverse direction reaches the input layer through a hidden layer, thus gradually modifying the connection weights, this algorithm is called “error back propagation algorithm” or “negative gradient correction algorithm”.

In this paper, a 3-5-1 network structure is considered, wherein the input layer consists of 3 nodes, the hidden layer consists of 5 nodes, and the output layer consists of 1 node. The detailed parameters are presented in

Table 1. BP neural network parameter settings.

Parameter Setting	Value
Number of input layer nodes	3
Number of hidden layer nodes	5
Number of output layer nodes	1
Number of training sessions	1000
Learning Rate	0.01
Minimum error of training target	0.001
Hidden layer transfer function	tansig
Hidden layer transfer function	purelin

and the combination is shown in Figure 3.

3.1.2. Markov and PSO

The particle swarm optimization (PSO) algorithm was first proposed by Eberhart and Kennedy in 1995 as a population-based stochastic optimization technique based on the foraging behavior of a flock of birds [33]. The PSO algorithm first initializes the particle states to obtain a set of random solutions and the particles continuously update their states by tracking individual local optimal solutions (Pbest) and global optimal solutions (Gbest) during spatial motion until the final global optimal solution is obtained by satisfying the conditions, such as the number of iterations or the error value.

In this paper, the Markov model is combined with PSO in a such a way as to obtain better predictions, mainly via training on how to select the position in the maximum probability interval. For example, the next data point predicted according to the Markov model has the highest probability of being located at (100, 120), the average value of this interval, 110, is taken as the final prediction result of the Markov model. The PSO is used to train the Markov model by taking 100 as the initial prediction value and adding the PSO training weights $\theta ·20$ as the final prediction value. The following is shown:

$$f=x+\theta ·w$$

(8)

where $f$ is the final prediction value, $x$ is the minimum value of the prediction interval of the Markov model, $\theta$ is the weight coefficient of the PSO algorithm training, and $w$ is the interval value of the Markov model.

As a result, the Markov model divides the interval $m$ if it is different and the corresponding number of $\theta$ is also different. In this experiment, $m$ was chosen to be 60. The PSO algorithm parameters were designed as follows: the search dimension is 60, the group size is 50, the number of evolutions is 100, the maximum particle velocity is 0.1, the minimum velocity is −0.1, the boundary is [−1, 1], and the learning factor $c_1=c_2=1.3$. The specific parameters of the PSO were set as shown in

Table 2. PSO parameter settings.

Parameter Setting	Value
Group size	50
Dimensionality	60
Number of evolutions	100
Acceleration constant c1	1.3
Acceleration constant c2	1.3
Maximum speed	0.1
Inertia weights	1.8

and the combination is shown in Figure 4. The fitness function was designed to minimize the sum of absolute values of errors between the predicted and actual values.

3.1.3. Markov Combined with PSO and BP Neural Network

The Markov-PSO-BP combination process is shown in Figure 5. The parameter settings of the PSO and BP neural networks are consistent with Table 1 and Table 2, respectively. The prediction results of the first-order Markov model are used as feature inputs and the first two steps of the original data are also used as inputs for a total of three levels of inputs. After that, PSO is used to optimize the weights of the BP neural network and the optimized results are fed back to the BP neural network, which is finally used to make the final prediction.

3.2. Evaluation Indicators

In this study, the performance of the model was evaluated based on the Mean Absolute Error (MAE), Root Mean Square Error (RMSE), Mean Square Error (MSE), Mean Absolute Percentage Error (MAPE), and the Coefficient of Determination (R²). The formulas for these metrics are shown below:

$$\mathrm{MAE}=\frac{1}{n}{\displaystyle \sum _{i=1}^n\left|(y_i-{\widehat{y}}_i)\right|} (\mathrm{MAE}\ge 0)$$

(9)

The Mean Absolute Error (MAE) is the average vertical distance between the actual and predicted values, where ${\widehat{y}}_i$ is the predicted value, $y_i$ is the actual value, and $n$ is the number of predictions.

$$\mathrm{MSE}=\frac{1}{n}{\displaystyle \sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2} (\mathrm{MSE}\ge 0)$$

(10)

Mean Square Error (MSE) is used to quantify the deviation between the predicted value and the cumulative probability distribution of the true value. A smaller MSE implies a smaller bias and better fitness.

$$\mathrm{RMSE}=\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2}} (\mathrm{RMSE}\ge 0)$$

(11)

Similar as in (9), a smaller RMSE indicates a better model effect.

$$\mathrm{R}-\mathrm{squared}=1-\frac{{\displaystyle \sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2}}{{\displaystyle \sum _{i=1}^n{(y_i-\overline{y})}^2}} (1\ge \mathrm{R}-\mathrm{squared}\ge 0)$$

(12)

where $\overline{y}$ is the mean value and $\mathrm{R}-\mathrm{squared}$ quantifies the probability distribution error between the predicted and true values. A larger $\mathrm{R}-\mathrm{squared}$ indicates a better fit of the predicted model to the true model.

$$\mathrm{MAPE}=\frac{1}{n}{\displaystyle \sum _{i=1}^n\left|\frac{{\widehat{y}}_i-y_i}{y_i}\right|}\times 100\% (\mathrm{MAPE}\ge 0)$$

(13)

The Mean Absolute Percentage Error (MAPE) is used to measure the error between the predicted value and the true value; the smaller the MAPE the better the model. Note that when the true value has data equal to 0, there is a denominator of 0 division issue and the formula is not available.

4. Numerical Experiments and Sensitivity Analysis

In this section, we conduct a series of comparative experiments to assess the performance and adaptability of the improved model. Our experimental setting is MATLAB2019a, Windows 10 Professional 64-bit, Intel Core i7-6700U CPU@2.60GHz, 8 GB RAM.

4.1. Data

Take the example of a wind farm in the United States run by a regional power transmission organization [34] with a wind power time interval of 1 h. The experiment used single-step prediction over a time period from 2 January 2017 to 2 January 2021; the total data volume was 35,064 data points.

In this study, we only consider wind power as an input variable, the reason being that meteorological parameters are not readily available at all locations. In addition, sharing Supervisory Control and Data Acquisition (SCADA) data violates the confidentiality policies of many companies.

The raw data were analyzed for any inaccuracies. Only 102 data points (out of 35,064 data points) were identified as missing or abnormal in the analysis using the quartile analysis method, which were subsequently imputed using the k-nearest neighbor algorithm.

4.2. CPU Time Comparison for the Improved Higher-Order Markov

We performed an experimental comparison between the mean prediction using the maximum probability interval on the probability distribution and the weighted mean using the probability interval, both of which can be found in the literature [30].

The parameters used in the experiment were described as follows: 800 data points were used for training data and 100 data points were used for the prediction data in dataset A. For dataset B, 5500 data points were used for training data and 500 data points were used for the prediction data. The number of interval splits used was 10–100. The outcomes of this experiment are presented in Figure 6.

The maximum probability interval approach outperforms the probability interval-weighted mean in prediction between the number of interval splits 10–30, while the probability interval-weighted mean performs better as the number of interval splits increases.

The prediction effect of the maximum probability interval is significantly better than the weighted mean of the probability interval when the number of interval splits is small (less than 50), and with the gradual increase of the probability interval, the prediction effect of the two tends to be similar except that the maximum probability interval in MAPE still has an advantage.

As mentioned earlier, a huge Transfer Probability Matrix (TPM) is generated in the conventional state classification implementing higher-order Markov. Our experiment with the second-order Markov model was carried out to show that with the increase in the number of segmentation intervals, the transition probability matrix becomes more and more complex.

The number of training data points was 33,000 and the number of prediction points was 1000 while changing the interval number from 10 to 200, as shown in Figure 7.

In Figure 7, it shows that the traditional model also tends to exponentially increase CPU Time as the number of interval partitions increases. According to the proposed algorithm in Section 2.2, our numerical experiments were carried out with traditional model parameters and those interval numbers ranged from 10 to 200, as shown in Figure 8.

From the experimental results in Figure 8, we can see that the CPU Time of the improved model experiences a significant decrease and does not increase with the number of partitions in the interval.

In addition, the comparison experiments between the first-order, second-order, and third-order Markov models and the modified first-order, second-order, and third-order Markov models were carried out, as shown in Figure 9 and Figure 10.

As shown in Figure 9, the CPU Time of the third-order Markov model is more pronounced after an exponential increase in the number of interval splits. In addition, in the first-, second-, and third-order Markov models with interval separation numbers of 10–100, the CPU Time of the first-order model shows a flat fluctuation with the number of interval splits, the CPU Time of the second-order model shows a fluctuating increase with the number of interval splits, and the CPU Time of the third-order model shows an exponential increase with the number of interval splits. As shown in Figure 10, the improved model showed little difference in the CPU Time for the first-, second-, and third-order models, and the increase in the number of interval splits did not significantly change the CPU Time. In addition, we summarized the experimental limitations and time complexity of the higher-order Markov model, shown in

Table 3. Conditional constraints of the higher-order Markov experiments.

Models	Number of Interval Splits	Prediction Points	Training Points	Transfer Probability Matrix Memory Space Size	Time Complexity
3rd-order Markov model	$m$	$pre\_num$	$train\_num$	$m^4$	$O_{(m^4+pre\_num+train\_num)}$
4th-order Markov model	$m$	$pre\_num$	$train\_num$	$m^5$	$O_{(m^5+pre\_num+train\_num)}$
5th-order Markov model	$m$	$pre\_num$	$train\_num$	$m^6$	$O_{(m^6+pre\_num+train\_num)}$
k-order Markov model	$m$	$pre\_num$	$train\_num$	$m^{k+1}$	$O_{(m^{k+1}+pre\_num+train\_num)}$
3rd-order improved Markov model	$m$	$pre\_num$	$train\_num$	N/A	$O_{(m\ast pre\_num)}$
4th-order improved Markov model	$m$	$pre\_num$	$train\_num$	N/A	$O_{(m\ast pre\_num)}$
5th-order improved Markov model	$m$	$pre\_num$	$train\_num$	N/A	$O_{(m\ast pre\_num)}$
k-order improved Markov model	$m$	$pre\_num$	$train\_num$	N/A	$O_{(m\ast pre\_num)}$

, Because the modified Markov model does not generate the transfer probability matrix, the size of the transfer matrix memory space is empty.

The memory space of the transfer probability matrix increases exponentially from the 3rd-order to k-order Markov models and the time complexity increases as well. In our experiments, the number of interval splits in the third-order Markov model could not exceed 181, the fourth-order model could not exceed 64, and the fifth-order model must be less than 33. The improved Markov model is not conditioned in this respect because there is no transfer probability matrix; also, its time complexity is lower and only increases linearly with the number of predicted points.

In addition, since a real-time calculation method was used, the number of predictions may have an impact on the CPU Time results, for which we have conducted corresponding experiments for analysis. We set the training data to 30,000 points, the number of intervals to 100, and the number of predictions from 10 to 1000 points.

Figure 11 shows that the increase in the number of predictions has little effect on the CPU Time of the conventional model, while Figure 12 illustrates that the improved model shows a linear increase with the increase in the number of predictions. Although the improved model increases linearly with the number of predictions, we found that it still requires less CPU Time, an advantage that should be even more pronounced in higher-order Markov models.

4.3. Comparison of Multiple Markov-Based Prediction Models

Using the method in Section 2.3, we conducted experiments on the weighted Markov model and also compared the first-order Markov-based prediction model, the second-order Markov-based prediction model, and the third-order Markov-based prediction model. The results are shown in Figure 13 and Figure 14, where MAE_1 (MSE_1, RMSE_1, R-squared_1, and MAPE_1) is determined by the 1st-order Markov model, MAE_2 (MSE_2, RMSE_2, R-squared_2, and MAPE_2) is determined by the 2nd-order Markov model, and MAE_3 (MSE_3, RMSE_3, R-squared_3, and MAPE_3) is determined by the 3rd-order Markov model.

From the experimental results in Figure 13, we can see that the prediction results of the first-, second-, and third-order Markov-based prediction models with a smaller amount of training data (800 points of training data) each win and lose with a smaller number of intervals, with little difference between them. However, when the number of intervals exceeds 100, the third-order Markov model generates a better prediction.

The prediction results shown in Figure 14 are the result of increasing the amount of training data points (5500 points of training data). The second-order Markov model performs better in the case of fewer orders. When the order exceeds 140, the third-order Markov model performs the best, the second-order Markov model is next, and the first-order model performs the worst. This result is consistent with the comparison results of the first- and second-order Markov models for wind speed prediction [35].

From the results in Figure 15, we can see that the first-order Markov model is significantly superior to the p-step weighted Markov model, and the larger the step, the worse the prediction effect, perhaps because of the increase in the number of distributions, which is also related to the initial weight setting of the weighted Markov model. By improving the setting of the weighted initial value, the prediction effect of the weighted Markov model may be improved.

We added a new evaluation indicator, SMAPE, which is defined as follows:

$$\mathrm{SMAPE}=\frac{1}{n}{\displaystyle \sum _{i=1}^n\frac{\left|{\widehat{y}}_i-y_i\right|}{\left(\left|y_i\right|+\left|\widehat{y}\right|\right)/2}}\times 100\% (SMAPE\ge 0)$$

(14)

The Symmetric Mean Absolute Percentage Error (SMAPE) is an accuracy measure based on percentage errors where ${\widehat{y}}_i$ is the predicted value, $y_i$ is the actual value, and $n$ is the number of predictions.

The Markov models for selecting the next moment prediction interval and the location of the interval points were improved based on the methods described in Section 3.1. The experiment utilized 1900 data points for training, 100 data points for prediction, and 60 interval splits. The results of the experiment are presented in

Table 4. Experimental comparison results of the improved Markov model.

Prediction Algorithms	MAE	MSE	RMSE	MAPE	R-squared	SMAPE	CPU Time
first-order Markov	238.51	97,269.98	311.88	17.81%	0.846	6.57%	0.359
second-order Markov	210.35	81,186.05	284.93	16.58%	0.871	5.31%	1.297
Markov-PSO-BP	179.26	64,441.47	253.85	12.7%	0.9005	4.44%	85.312
Markov-PSO	237.28	94,774.67	307.85	17.76%	0.8537	6.49%	4.312
ARIMA(2,0,2)	183.34	65,783.07	256.48	13.58%	0.9116	4.56%	9.203
BPNN	185.01	67,882.95	260.54	13.12%	0.8952	4.63%	0.515
Markov-BP	183.62	66,599.47	258.06	12.96%	0.8972	4.61%	0.687
two-step weighted Markov	279	135,322.8	367.85	20.91%	0.7911	7.21%	0.562

.

Based on the results shown in Table 4, the Markov-PSO-BP combination has the best prediction effect, which indicates that the combined Markov-PSO-BP model proposed in this paper has high prediction accuracy and is closer to the actual wind power output. The predicted values show the best performance in evaluation indices such as MAE, MES, RMSE, MAPE, and SMAPE, while the Coefficient of Determination is slightly inferior to that of the ARIMA model. On the other hand, the two-step weighted Markov has the worst prediction effect. The combination of Markov and PSO also improves the prediction of the original model, but the improvement is not very significant and is only slightly better than the first-order Markov model. Moreover, the first- and second-order Markov models show poor prediction effects compared to the ARIMA and BP neural network models.

Specifically, the ARIMA model has an advantage in the R-squared metric, while the first-order Markov model takes the shortest amount of CPU Time, and the combined Markov-PSO-BP model requires more time to adjust the parameters for prediction. In the future, we can consider making improvements in this trend to improve the prediction effect of the Markov model while accounting for environmental and geographical factors that influence short-term wind power and further explore the information characteristics of short-term wind power. Furthermore, doing this will also build more accurate short-term wind power prediction models.

5. Conclusions

In this paper, an algorithm for improving the k-order Markov model was proposed, which addresses the challenge of lengthy CPU Time caused by the complexity of the traditional model, thus making higher-order Markov models more applicable. From those illustrations in the numerical experiments, we conclude that the proposed k-order Markov-based prediction algorithm can provide faster computation while requiring less memory space and retain the same level of prediction accuracy as compared to the traditional models; further, this effect will be more prominent in higher-order models.

When employing Markov-based models for point prediction, there are several methods for selecting the probability interval. Among them, the maximum probability selection method has a better prediction effect compared with mean selection when dealing with a larger data sample size. Additionally, optimizing the position of the maximum probability interval can further enhance the model’s prediction capability, which is not applicable to the mean value model. In this paper, we integrated the Markov model with PSO-BP to improve the accuracy of point prediction. By utilizing the PSO algorithm to optimize the network weights of the BP neural network, we enhanced its global optimization ability and leverage the adaptive and learning capability of the BP neural network. In terms of probability interval selection, a smaller probability interval selection leads to superior model predictions, as the finer the probability interval segment, the closer the final predicted value is to the true value of the next moment. From the numerical results derived in several examples, it can be observed that the presented method performs better in terms of prediction accuracy and stability.

In future work, we anticipate that the improved k-order Markov-based real-time algorithm may be applied to other fields, such as combining it with Monte Carlo simulations of wind power output. The weight optimization of the studied PSO-BP neural network could be further improved via other intelligent optimization algorithms, such as the Whale Optimization Algorithm [36], Wolf Pack Algorithm [37], Dragonfly Algorithm, Ant Lion Algorithm, and so on. Besides, in this paper, only wind power is considered as an input variable. In the future, we anticipate that geo-environmental information, including temperature, humidity, and other relevant data, can be effectively utilized as input features for predictive modeling. This approach will involve the use of various analytical techniques, such as principal component analysis, to analyze and process the multidimensional data obtained from these information sources [38,39].